本文關(guān)鍵詞：相空間中非保守系統(tǒng)的Herglotz廣義變分原理及其Noether定理　出處：《力學(xué)學(xué)報》2016年06期 　論文類型：期刊論文

  更多相關(guān)文章： Herglotz廣義變分原理 Noether定理 非保守動力學(xué) 相空間

【摘要】：與經(jīng)典變分原理相比,基于由微分方程定義的作用量的Herglotz廣義變分原理給出了非保守動力學(xué)系統(tǒng)的一個變分描述,它不僅能夠描述所有采用經(jīng)典變分原理能夠描述的動力學(xué)過程,而且能夠應(yīng)用于經(jīng)典變分原理不能適用的非保守或耗散系統(tǒng).將Herglotz廣義變分原理拓展到相空間,研究相空間中非保守力學(xué)系統(tǒng)的Herglotz廣義變分原理與Noether定理及其逆定理.首先,提出相空間中Herglotz廣義變分原理,給出相空間中非保守系統(tǒng)的變分描述,導(dǎo)出相應(yīng)的Hamilton正則方程;其次,基于非等時變分與等時變分之間的關(guān)系,導(dǎo)出相空間中Hamilton-Herglotz作用量變分的兩個基本公式;再次,給出Noether對稱變換的定義和判據(jù),提出并證明相空間中非保守系統(tǒng)基于Herglotz變分問題的Noether定理及其逆定理,揭示了相空間中力學(xué)系統(tǒng)的Noether對稱性與守恒量之間的內(nèi)在聯(lián)系.在經(jīng)典條件下,Herglotz廣義變分原理退化為經(jīng)典變分原理,與之相應(yīng)的相空間中的Noether定理退化為經(jīng)典Hamilton系統(tǒng)的Noether定理.文末以著名的Emden方程和平方阻尼振子為例說明上述方法和結(jié)果的有效性.
[Abstract]:Compared with the classical variational principle, based on the Herglotz generalized variational principle defined by the differential equation, a variational description of a nonconservative dynamical system is given. It can not only describe all the dynamic processes that can be described by the classical variational principle. Moreover, it can be applied to non-conservative or dissipative systems where the classical variational principle is not applicable. The Herglotz generalized variational principle is extended to the phase space. The Herglotz generalized variational principle and Noether theorem and their inverse theorems for non-conservative mechanical systems in phase space are studied. Firstly, the Herglotz generalized variational principle in phase space is proposed. The variational description of non-conservative systems in phase space is given, and the corresponding Hamilton canonical equations are derived. Secondly, based on the relationship between non-equal-time-varying and equal-time-varying, two basic formulas of Hamilton-Herglotz action variation in phase space are derived. Thirdly, the definition and criterion of Noether symmetry transformation are given, and the Noether theorem and its inverse theorem for non-conservative systems in phase space based on Herglotz variational problem are presented and proved. The relationship between Noether symmetry and conserved quantities of mechanical systems in phase space is revealed. Under classical conditions, Herglotz generalized variational principle degenerates into classical variational principle. The corresponding Noether theorem in phase space degenerates to the Noether theorem of classical Hamilton system. At the end of this paper, the famous Emden equation and squared damped oscillator are taken as examples. The validity of the method and result is described.
【作者單位】： 蘇州科技大學(xué)土木工程學(xué)院;
【基金】：國家自然科學(xué)基金資助項(xiàng)目(11272227,11572212)
【分類號】：O316
【正文快照】： 引言 Noether定理首次從變分學(xué)背景下揭示了對稱性與守恒量之間的相互關(guān)系[1],即Hamilton作用量在關(guān)于廣義坐標(biāo)和時間的變換群的無限小變換下的不變性意味著沿著系統(tǒng)的動力學(xué)真實(shí)運(yùn)動軌道存在一個守恒量.Noether定理闡釋了牛頓力學(xué)的所有守恒量,如:時間的均R緣賈輪實(shí)閬檔，

本文編號：1373734